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My friend has a green Infect deck. When he declares a creature to attack and I allow it to go through unblocked, he suddenly plays all these Instants that 'pump' the creature up, and because of Infect, it's an instant loss most of the time. We solved that issue (without really understanding the full rules) that it wouldn't make sense to pump his creature up for an instant win without allowing me to block, although recently, my friend told me that 'Instants' can be played 'anytime' so once I allow a creature to deal me damage, technically, he's allowed to play all those Instants.

I'm really confused about it all, and would like to get an answer once and for all for this.

Also, once he has stacked all the pumps for his creature and attacks me (after I initially let it pass to hit my life because I did not expect the pumps) am I allowed to play Holy Day?

At the end of each step (except untap and cleanup*), any abilities that trigger at the start of that step are put on the stack, and then the active player (whoever's turn it is) receives priority.

Priority means you can cast spells, activate abilities or play a land. (Keep in mind that casting non-instant spells, activating some abilities and playing lands can only be done if it's your turn, if it's a Main phase, and if the stack is empty.) If a player is done, they may pass priority. Then, the next player gets priority to play spells and activate abilities.

When all players pass priority, any spells on the stack resolve. After each spell or ability resolves the active player receives priority again. He may play new fast effects (instant spells or activate abilities.)

If the stack is empty and all players pass priority, the game moves to the next step or phase.

* — Players don't receive priority during the Untap or Cleanup Step, with one exception during the Cleanup Step. Check the comprehensive rules 514 for more info.

The Stack

The stack is a holding area for spells and abilities that have been cast or activated (including pumping a creature or playing a spell like Holy Day.) After a spell has been played or an ability activated, it is put on the stack.

Once everyone has passed priority, the last-played spell or ability on the stack resolves. It's like a stack of dishes or paper. You deal with the one on top first. After each spell or ability resolves, the active player receives priority again. When all players pass, the next spell on the stack resolves, and so on, until the stack is empty.

Conclusion

You always have a chance to respond to your opponent casting spells and activating abilities to pump his creature (ignoring Split Second.) And your opponent can choose to pump his creature during phase 1 & 2, but usually it is most advantageous to do it during the Declare Blockers step of the Combat phase, after blockers are declared.

I hope that makes sense. I've tried to boil it down "When can I play instants?" to the essentials and a minimum of caveats and exceptions. Magic has many card interactions and rules to deal with these nitty-gritty details; browse the comprehensive rules if you really want to get into those!

I'm going to say the same thing as the other answers, but more focused on your specific case.

As has been said elsewhere, here's what happens during combat:

Begin combat step

Both players get to play spells and abilities

Declare attackers step

The attacking player chooses which creatures are attacking. Once the choices are made, they are locked in.

Both players get to play spells and abilities

Declare blockers step

The defending player chooses which creatures are blocking. Once the choices are made, they are locked in; in particular, if you choose not to block a creature, you can't later go back and block it .

Both players get to play spells and abilities. This is where your infect-playing opponent will play his pump spells.

Combat damage step

Combat damage is dealt

Both players get to play spells and abilities

End of combat step

Both players get to play spells and abilities

When I say "Both players get to play spells and abilities," that's code for "each player gets priority" and all that other stuff that ghoppe explained. I'm not going to go into that., except to reiterate that after he plays a spell, you will also get a chance to play spells, and vice versa. The game doesn't move on to the next step until both of you decline to play anything.

So the bottom line is that yes, your opponent can play spells to pump up his creature after you block or decline to block. And you can play spells like Holy Day to prevent the damage those creatures would do, or whatever other spells you want to play.

Now you may be thinking that this is kind of unfair, because he can attack with a small creature, which you don't block because it doesn't seem threatening, and then he can pump it up to instantly kill you. But it's not. Once you see your opponent pull this trick once, you know about it, and next time, you can block his creature, even if it's small, because you know that if you don't ,he will be able to pump it up and kill you. Consider this your introduction to the strategy of Magic: it's about learning to anticipate what your opponent could play before he does it. You can do this based on what you know to be in his deck, how much mana he has available, how many cards he has in hand, and any other information that might be relevant.

Actually, playing against an infect deck is one of the easier matchups to play, strategically speaking, because all you have to do is block everything whenever your opponent might be able to play a pump spell. Remember that infect creatures are small for their cost, so when you block, either your blocker will kill the infect creature, or your opponent will have to use a pump spell to save it. After 3 or 4 blocks, your opponent will either be out of pump spells (in which case his creatures aren't threatening anymore) or out of creatures (in which case the pump spells are useless).

Players have a chance to cast instants and activate abilities at the end of each of those steps, including after he declares attackers and you declare blockers. After he casts instants you are also given a chance to casts instants.

Your opponent is allowed to pump up his creatures after you decline to block, resulting in you taking the damage, or you playing a spell or an ability to prevent the damage or kill that creature.

Let's say your opponent has a simple 1/1 and swings for 1; you don't block. Because you do not block, he can now pump his creature up and swing at you for X. You seeing this can now play a Holy Day, Lightning Bolt, Incinerate, etc. to prevent the damage or kill the creature. These spells are now on the stack, and because yours was the last spell played, it resolves first. He pumps his 1/1 up to a 4/4, and in response, you use Shock to kill it before it becomes a 4/4. Because you did not block during declare blockers phase, you can't block now, but you still have the opportunity to cast instants/activate abilities.

Yes, he can play those instants after you declined to block. That's one of the most useful things to do with instant spells, and is done a lot after blockers are announced. Keep in mind about the stacking: if he stacks an instant on top of yours, it may well counteract what you're trying to do.